On the dimension of self-similar measures with complicated overlaps

Super-exponential condensation without exact overlaps

We exhibit self-similar sets on the line which are not exponentially separated and do not generate any exact overlaps. Our result shows that the exponential separation, introduced by Hochman in his groundbreaking theorem on the dimension of self-similar sets, is too weak to describe the full theory.

Projection Pressure and Bowen's Equation for a Class of Self-similar Fractals with Overlap Structure

Let $\{S_i\}_{i=1}^{l}$ be an iterated function system(IFS) on $\mathbb{R}^d$ with attractor K. Let $π$ be the canonical projection. In this paper we define a new concept called "projection pressure" $P_π(ϕ)$ for $ϕ\in C(\mathbb{R}^d)$ under certain affine IFS, and show the variational principle about the projection pressure. Furthermore we check that the unique zero root of "projection pressure" still satisfies Bowen's equation when each $S_i$ is the similar map with the same compression ratio. Using the root of Bowen's equation, we can get the Hausdorff dimension of the attractor $K$.

实数直线上自相似测度的精确重叠猜想

The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line

本文证明了一维自相似测度的精确重叠猜想及其推广，通过新熵不等式量化信息损失，建立维数与随机游走熵及Lyapunov指数的联系。

Lipschitz equivalence of self-similar sets with exact overlaps

In this paper, we study a class $\mathcal{A}(λ,n,m)$ of self-similar sets with $m$ exact overlaps generated by $n$ similitudes of the same ratio $ λ$. We obtain a necessary condition for a self-similar set in $\mathcal{A}(λ,n,m)$ to be Lipschitz equivalent to a self-similar set satisfying the strong separation condition, i.e., there exists an integer $ k\geq 2$ such that $x^{2k}-mx^{k}+n$ is reducible, in particular, $m$ belongs to $\{a^{i}:a\in \mathbb{N}$ with $i\geq 2\}.$

Dirichlet forms on self-similar sets with overlaps

We study Dirichlet forms and Laplacians on self-similar sets with overlaps. A notion of "finitely ramified of finite type($f.r.f.t.$) nested structure" for self-similar sets is introduced. It allows us to reconstruct a class of self-similar sets in a graph-directed manner by a modified setup of Mauldin and Williams, which satisfies the property of finite ramification. This makes it possible to extend the technique developed by Kigami for analysis on $p.c.f.$ self-similar sets to this more general framework. Some basic properties related to $f.r.f.t.$ nested structures are investigated. Several…

Inhomogeneous self-similar sets with overlaps

It is known that if the underlying iterated function system satisfies the open set condition, then the upper box dimension of an inhomogeneous self-similar set is the maximum of the upper box dimensions of the homogeneous counterpart and the condensation set. First, we prove that this `expected formula' does not hold in general if there are overlaps in the construction. We demonstrate this via two different types of counterexample: the first is a family of overlapping inhomogeneous self-similar sets based upon Bernoulli convolutions; and the second applies in higher dimensions and makes use of…

A Multifractal Decomposition for Self-similar Measures with Exact Overlaps

We study self-similar measures in $\mathbb{R}$ satisfying the weak separation condition along with weak technical assumptions which are satisfied in all known examples. For such a measure $μ$, we show that there is a finite set of concave functions $\{τ_1,\ldots,τ_m\}$ such that the $L^q$-spectrum of $μ$ is given by $\min\{τ_1,\ldots,τ_m\}$ and the multifractal spectrum of $μ$ is given by $\max\{τ_1^*,\ldots,τ_m^*\}$, where $τ_i^*$ denotes the concave conjugate of $τ_i$. In particular, the measure $μ$ satisfies the multifractal formalism if and only if its multifractal spectrum is a concave fu…

On multifractal formalism for self-similar measures with overlaps

Let $μ$ be a self-similar measure generated by an IFS $Φ=\{ϕ_i\}_{i=1}^\ell$ of similarities on $\mathbb R^d$ ($d\ge 1$). When $Φ$ is dimensional regular (see Definition~1.1), we give an explicit formula for the $L^q$-spectrum $τ_μ(q)$ of $μ$ over $[0,1]$, and show that $τ_μ$ is differentiable over $(0,1]$ and the multifractal formalism holds for $μ$ at any $α\in [τ_μ'(1),τ_μ'(0+)]$. We also verify the validity of the multifractal formalism of $μ$ over $[τ_μ'(\infty),τ_μ'(0+)]$ for two new classes of overlapping algebraic IFSs by showing that the asymptotically weak separation condition holds.…

On a kind of self-similar sets with complete overlaps

Let $E$ be the self-similar set generated by the {\it iterated function system} {\[ f_0(x)=\frac{x}β,\quad f_1(x)=\frac{x+1}β, \quad f_{β+1}=\frac{x+β+1}β \]}with $β\ge 3$. {Then} $E$ is a self-similar set with complete {overlaps}, i.e., $f_{0}\circ f_{β+1}=f_{1}\circ f_1$, but $E$ is not totally self-similar. We investigate all its generating iterated function systems, give the spectrum of $E$, and determine the Hausdorff dimension and Hausdorff measure of $E$ and of the sets which contain all points in $E$ having finite or infinite different triadic codings.

Multiplication on self-similar sets with overlaps

Let $A,B\subset\mathbb{R}$. Define $$A\cdot B=\{x\cdot y:x\in A, y\in B\}.$$ In this paper, we consider the following class of self-similar sets with overlaps. Let $K$ be the attractor of the IFS $\{f_1(x)=λx, f_2(x)=λx+c-λ,f_3(x)=λx+1-λ\}$, where $f_1(I)\cap f_2(I)\neq \emptyset, (f_1(I)\cup f_2(I))\cap f_3(I)=\emptyset,$ and $I=[0,1]$ is the convex hull of $K$. The main result of this paper is $K\cdot K=[0,1]$ if and only if $(1-λ)^2\leq c$. Equivalently, we give a necessary and sufficient condition such that for any $u\in[0,1]$, $u=x\cdot y$, where $x,y\in K$.

On the structure of $λ$-Cantor set with overlaps

Given $λ\in(0, 1)$, let $E_λ$ be the self-similar set generated by the iterated function system $\{x/3,(x+λ)/3,(x+2)/3\}$. Then $E_λ$ is a self-similar set with overlaps. We obtain the necessary and sufficient condition for $E_λ$ to be totally self-similar, which is a concept first introduced by Broomhead, Montaldi, and Sidorov in 2004. When $E_λ$ is totally self-similar, all its generating IFSs are investigated, and the size of the set of points having finite triadic codings is determined. Besides, we give some properties of the spectrum of $E_λ$ and show that the spectrum of $E_λ$ vanishes i…

从重叠迭代函数系统的顶端生成铺砌

Tilings from Tops of Overlapping Iterated Function Systems

本文研究了如何通过重叠迭代函数系统的顶端扩展自相似铺砌，提出了一种收敛于铺砌的近似超瓷砖序列的方法，并探讨了有限类型系统、非周期性单体铺砌及无限多种相关原型瓷砖的实例。

On the Assouad dimension of self-similar sets with overlaps

It is known that, unlike the Hausdorff dimension, the Assouad dimension of a self-similar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for self-similar sets in the line: either the \emph{weak separation property} is satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the \emph{weak separation property} is not satisfied, in which case the Assouad dimension is maximal (equal to one). In the first case we prove that the self-similar set is Ahlfors regular, and in the second case we use …

Estimates on the dimension of self-similar measures with overlaps

In this paper, we provide an algorithm to estimate from below the dimension of self-similar measures with overlaps. As an application, we show that for any $ β\in(1,2) $, the dimension of the Bernoulli convolution $ μ_β$ satisfies \[ \dim (μ_β) \geq 0.9804085,\] which improves a previous uniform lower bound $0.82$ obtained by Hare and Sidorov \cite{HareSidorov2018}. This new uniform lower bound is very close to the known numerical approximation $ 0.98040931953\pm 10^{-11}$ for $\dim μ_{β_3}$, where $ β_{3} \approx 1.839286755214161$ is the largest root of the polynomial $ x^{3}-x^{2}-x-1$. Mor…

带重叠的自相似测度的加倍性质

Doubling property of self-similar measures with overlaps

本文针对满足特定Pisot数方程的带重叠自相似测度，拓展了已有开集条件下的加倍性研究，证明当参数m≥3时该类测度恒非加倍，补充了m=2时的已知结论。

Assouad dimension and local structure of self-similar sets with overlaps in $\mathbb{R}^d$

For a self-similar set in $\mathbb{R}^d$ that is the attractor of an iterated function system that does not verify the weak separation property, Fraser, Henderson, Olson and Robinson showed that its Assouad dimension is at least $1$. In this paper, it is shown that the Assouad dimension of such a set is the sum of the dimension of the vector space spanned by the set of $\textit{overlapping directions}$ and the Assouad dimension of the orthogonal projection of the set self-similar set onto the orthogonal complement of that vector space. This result is applied to give sufficient conditions on th…

一族随机自相似迭代函数系统的维数性质

Dimension properties of a family of random self-similar iterated function systems

本文针对一类具有良好组织重叠结构的随机自相似集，基于期望矩阵的压力函数给出了其维数公式，并推广了非均匀海绵体的有理投影。

Multiple codings for self-similar sets with overlaps

In this paper we consider a general class $\mathcal E$ of self-similar sets with complete overlaps. Given a self-similar iterated function system $Φ=(E, \{f_i\}_{i=1}^m)\in\mathcal E$ on the real line, for each point $x\in E$ we can find a sequence $(i_k)=i_1i_2\ldots\in\{1,\ldots,m\}^\mathbb N$, called a coding of $x$, such that $$ x=\lim_{n\to\infty}f_{i_1}\circ f_{i_{2}}\circ\cdots\circ f_{i_n}(0). $$ For $k=1,2,\ldots, \aleph_0$ or $2^{\aleph_0}$ we investigate the subset $\mathcal U_k(Φ)$ which consists of all $x\in E$ having precisely $k$ different codings. Among several equivalent chara…

Intrinsic Diophantine Approximation for overlapping iterated function systems

In this paper we study a family of limsup sets that are defined using iterated function systems. Our main result is an analogue of Khintchine's theorem for these sets. We then apply this result to the topic of intrinsic Diophantine Approximation on self-similar sets. In particular, we define a new height function for an element of $\mathbb{Q}^d$ contained in a self-similar set in terms of its eventually periodic representations. For limsup sets defined with respect to this height function, we obtain a detailed description of their metric properties. The results of this paper hold in arbitrary …

Quasi-doubling of self-similar measures with overlaps

The Assouad and quasi-Assouad dimensions of a metric space provide information about the extreme local geometric nature of the set. The Assouad dimension of a set has a measure theoretic analogue, which is also known as the upper regularity dimension. One reason for the interest in this notion is that a measure has finite Assouad dimension if and only if it is doubling. Motivated by recent progress on both the Assouad dimension of measures that satisfy a strong separation condition and the quasi-Assouad dimension of metric spaces, we introduce the notion of the quasi-Assouad dimension of a mea…

On self-similar sets with overlaps and inverse theorems for entropy

We study the Hausdorff dimension of self-similar sets and measures on the line. We show that if the dimension is smaller than the minimum of 1 and the similarity dimension, then at small scales there are super-exponentially close cylinders. This is a step towards the folklore conjecture that such a drop in dimension is explained only by exact overlaps, and confirms the conjecture in cases where the contraction parameters are algebraic. It also gives an affirmative answer to a conjecture of Furstenberg, showing that the projections of the "1-dimensional Sierpinski gasket" in irrational directio…

Lipschitz equivalence of a class of self-similar sets

We consider a class of homogeneous self-similar sets with complete overlaps and give a sufficient condition for the Lipschitz equivalence between members in this class.

Dimensions of an overlapping generalization of Barański carpets

We determine the Hausdorff, packing and box-counting dimension of a family of self-affine sets generalizing Barański carpets. More specifically, we fix a Barański system and allow both vertical and horizontal random translations, while preserving the rows and columns structure. The alignment kept in the construction allows us to give expressions for these fractal dimensions outside of a small set of exceptional translations. Such formulas will coincide with those for the non-overlapping case, and thus provide examples where the box-counting and Hausdorff dimension do not necessarily agree. The…

On the dimensions of attractors of random self-similar graph directed iterated function systems

In this paper we propose a new model of random graph directed fractals that extends the current well-known model of random graph directed iterated function systems, $V$-variable attractors, and fractal and Mandelbrot percolation. We study its dimensional properties for similarities with and without overlaps. In particular we show that for the two classes of $1$-variable and $\infty$-variable random graph directed attractors we introduce, the Hausdorff and upper box counting dimension coincide almost surely, irrespective of overlap. Under the additional assumption of the uniform strong separati…

Multiple representations of real numbers on self-similar sets with overlaps

Let $K$ be the attractor of the following IFS $$\{f_1(x)=λx, f_2(x)=λx +c-λ,f_3(x)=λx +1-λ\}, $$ where $f_1(I)\cap f_2(I)\neq \emptyset, (f_1(I)\cup f_2(I))\cap f_3(I)=\emptyset,$ and $I=[0,1]$ is the convex hull of $K$. The main results of this paper are as follows: $$\sqrt{K}+\sqrt{K}=[0,2]$$ if and only if $$\sqrt{c}+1\geq 2\sqrt{1-λ},$$ where $\sqrt{K}+\sqrt{K}=\{\sqrt{x}+\sqrt{y}:x,y\in K\}$. If $c\geq (1-λ)^2$, then $$\dfrac{K}{K}=\left\{\dfrac{x}{y}:x,y\in K, y\neq 0\right\}=\left[0,\infty\right).$$ As a consequence, we prove that the following conditions are equivalent: (1) For any $u\…

On the dimensions of a family of overlapping self-affine carpets

We consider the dimensions of a family of self-affine sets related to the Bedford-McMullen carpets. In particular, we fix a Bedford-McMullen system and then randomise the translation vectors with the stipulation that the column structure is preserved. As such, we maintain one of the key features in the Bedford-McMullen set up in that alignment causes the dimensions to drop from the affinity dimension. We compute the Hausdorff, packing and box dimensions outside of a small set of exceptional translations, and also for some explicit translations even in the presence of overlapping. Our results r…

Overlapping iterated function systems from the perspective of Metric Number Theory

In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation. This result shows that for a family of limsup sets, their Lebesgue measure is determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems our results apply to include those arising from Bernoulli convol…